교과서 질문
In Exercises 25–29, use DeMoivre's Theorem to find the indicated power of the complex number. Write answers in rectangular form. [2(cos 20° + i sin 20°)]³
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Table of contents
5장, 문제 25
In Exercises 21–40, eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that −∞ < t < ∞. _ x = √t, y = t − 1
검증된 단계별 안내1
Start with the given parametric equations: \(x = \sqrt{t}\) and \(y = t - 1\).
Express \(t\) in terms of \(x\) from the first equation: since \(x = \sqrt{t}\), then \(t = x^2\) (note that \(x \geq 0\) because square roots are non-negative).
Substitute \(t = x^2\) into the second equation to eliminate the parameter: \(y = x^2 - 1\).
Recognize that the rectangular equation is \(y = x^2 - 1\), which is a parabola opening upwards, shifted down by 1 unit.
To sketch the curve, plot the parabola \(y = x^2 - 1\) for \(x \geq 0\) (since \(x = \sqrt{t}\) implies \(x \geq 0\)), and use arrows pointing in the direction of increasing \(t\) (which corresponds to increasing \(x\) and \(y\) values).
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주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Parametric Equations and Parameters
Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted as t. Understanding how the parameter t controls the position on the curve is essential for analyzing the curve's shape and orientation.
추천 영상:
05:59
Eliminating the Parameter
Eliminating the Parameter
Eliminating the parameter involves rewriting the parametric equations to form a single rectangular equation in terms of x and y. This process helps to identify the curve's equation in the Cartesian plane, making it easier to analyze and sketch.
추천 영상:
05:59Eliminating the Parameter
Sketching and Orientation of Curves
Sketching the curve requires plotting points that satisfy the rectangular equation and indicating the direction of increasing parameter t with arrows. Orientation shows how the curve is traced as t increases, which is important for understanding the curve's behavior.
추천 영상:
06:05Eliminate the Parameter Example 2
관련 실천
교과서 질문
In Exercises 11–26, plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. −3 + 4i
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교과서 질문
In Exercises 13–34, test for symmetry and then graph each polar equation. r = 4 sin 3θ
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교과서 질문
In Exercises 27–36, write each complex number in rectangular form. If necessary, round to the nearest tenth. 6(cos 30° + i sin 30°)
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교과서 질문
In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i
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교과서 질문
In Exercises 21–26, use a polar coordinate system like the one shown for Exercises 1–10 to plot each point with the given polar coordinates. Then find another representation of this point in which a. r>0, 2π < θ < 4π. b. r<0, 0. < θ < 2π. c. r>0, −2π. < θ < 0. (4, π/2)
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